Group #31902262 — subgroups

Subgroups — #31902262

Order 18176.

294 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 87 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22588481	C₂	1	1	yes	maximal, characteristic
2	#22587012	C₂	1	2	yes	maximal
3	#22586198	C₂	1	2	yes	maximal
4	#22586185	C₂	1	1	yes	maximal, characteristic
5	C₇₁×#1960	C₂	1	1	yes	maximal, characteristic
6	#17986052	C₂²	1	2	yes	—
7	#17986023	C₂²	1	2	yes	—
8	#17985764	—	2	2	no	—
9	#17985745	—	2	4	no	—
10	C₂×C₄⋊Q₈×C₇₁	C₂²	1	1	yes	characteristic
11	C₂×#103×C₇₁	—	2	2	no	—
12	C₂×#97×C₇₁	C₂²	1	1	yes	characteristic
13	#382×C₇₁	C₂²	1	1	yes	characteristic
14	C₂×#11765849	—	2	2	no	—
15	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
16	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
17	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
18	C₂²⋊(C₇₁⋊C₈)	—	2	2	no	—
19	C₄×C₇₁⋊C₈	—	2	2	no	—
20	C₂²×Q₈×C₇₁	—	2	1	no	—
21	C₂²×C₈×C₇₁	D₄	1	2	yes	—
22	C₄⋊Q₈×C₇₁	—	2	2	no	—
23	C₄⋊Q₈×C₇₁	—	2	1	no	—
24	C₄⋊Q₈×C₇₁	D₄	1	2	yes	—
25	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
26	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
27	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
28	#103×C₇₁	—	4	4	no	—
29	#97×C₇₁	—	2	2	no	—
30	#95×C₇₁	—	2	2	no	—
31	#11765849	—	4	2	no	—
32	#11765849	—	4	2	no	—
33	C₂×C₇₁⋊C₈	—	4	2	no	—
34	C₂×C₇₁⋊C₈	—	4	2	no	—
35	C₂×Q₈×C₇₁	—	2	2	no	—
36	C₂×Q₈×C₇₁	—	4	1	no	—
37	C₂×Q₈×C₇₁	—	4	2	no	—
38	C₂×Q₈×C₇₁	—	4	2	no	—
39	C₂²×C₄×C₇₁	—	4	1	no	—
40	C₂²×C₄×C₇₁	—	4	1	no	—
41	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₇₁	#41	1	2	yes	—
43	C₂×C₈×C₇₁	—	2	2	no	—
44	C₂×C₈×C₇₁	—	2	2	no	—
45	C₂×C₈×C₇₁	—	2	2	no	—
46	C₄⋊C₄×C₇₁	—	2	4	no	—
47	C₄⋊C₄×C₇₁	—	4	2	no	—
48	C₄⋊C₄×C₇₁	—	4	2	no	—
49	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
50	C₄²×C₇₁	#41	1	2	yes	—
51	C₄²×C₇₁	#41	1	1	yes	characteristic
52	C₂³×C₇₁	#124	1	1	yes	characteristic
53	Q₈×C₇₁	—	4	4	no	—
54	Q₈×C₇₁	—	8	2	no	—
55	C₂×C₄×C₇₁	—	4	2	no	—
56	C₂×C₄×C₇₁	—	4	2	no	—
57	C₂×C₄×C₇₁	—	8	1	no	—
58	C₂×C₄×C₇₁	—	8	1	no	—
59	C₂×C₄×C₇₁	—	8	2	no	—
60	C₂×C₄×C₇₁	—	8	1	no	—
61	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
62	C₂×C₄×C₇₁	#123	1	1	yes	characteristic
63	C₂×C₄×C₇₁	#123	1	2	yes	—
64	C₂×C₄×C₇₁	#126	1	2	yes	—
65	C₂×C₄×C₇₁	#135	1	2	yes	—
66	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	2	yes	—
67	C₈×C₇₁	—	4	2	no	—
68	C₇₁⋊C₈	—	8	2	no	—
69	C₇₁⋊C₈	—	8	2	no	—
70	C₂²×C₇₁	—	2	2	no	—
71	C₂²×C₇₁	#397	1	1	yes	characteristic
72	C₂²×C₇₁	#488	1	1	yes	characteristic
73	C₂²×C₇₁	#490	1	1	yes	characteristic
74	C₄×C₇₁	—	2	2	no	—
75	C₄×C₇₁	—	2	2	no	—
76	C₄×C₇₁	—	8	2	no	—
77	C₄×C₇₁	—	16	1	no	—
78	C₄×C₇₁	#463	1	2	yes	—
79	C₄×C₇₁	#482	1	2	yes	—
80	#10635	—	71	1	no	maximal, Sylow
81	C₂×C₇₁	—	2	1	no	—
82	C₂×C₇₁	—	2	1	no	—
83	C₂×C₇₁	#2073	1	1	yes	characteristic
84	C₂×C₇₁	#2082	1	2	yes	—
85	#1960	D₇₁	1	1	yes	characteristic
86	#1838	—	71	1	no	—
87	#1546	—	71	2	no	—
88	#1286	—	71	2	no	—
89	#1281	—	71	1	no	—
90	C₇₁	#10635	1	1	yes	characteristic, complemented, Sylow
91	C₂×C₄⋊Q₈	C₂×D₇₁	1	1	yes	characteristic
92	#431	—	71	2	no	—
93	#423	—	71	2	no	—
94	C₂×#103	—	2	2	no	—
95	C₂×#97	C₂×D₇₁	1	1	yes	characteristic
96	#382	C₂×D₇₁	1	1	yes	characteristic
97	#341	—	142	2	no	—
98	#332	—	142	4	no	—
99	C₂²×Q₈	—	2	1	no	—
100	C₂×M₁₆	—	142	2	no	—
101	C₂²×C₈	C₂²⋊D₇₁	1	2	yes	—
102	C₄⋊Q₈	—	2	2	no	—
103	C₄⋊Q₈	—	2	1	no	—
104	C₄⋊Q₈	D₂₈₄	1	2	yes	—
105	C₂×C₄⋊C₄	—	2	2	no	—
106	C₂×C₄⋊C₄	—	2	1	no	—
107	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
108	C₄⋊C₈	—	142	2	no	—
109	C₄⋊C₈	—	142	2	no	—
110	C₄⋊C₈	—	142	2	no	—
111	#103	—	4	4	no	—
112	C₂²⋊C₈	—	142	2	no	—
113	#97	—	2	2	no	—
114	C₄×C₈	—	142	2	no	—
115	#95	—	2	2	no	—
116	C₂×Q₈	—	2	2	no	—
117	C₂×Q₈	—	4	1	no	—
118	C₂×Q₈	—	4	2	no	—
119	C₂×Q₈	—	4	2	no	—
120	C₂²×C₄	—	4	1	no	—
121	C₂²×C₄	—	4	1	no	—
122	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
123	C₂²×C₄	#11765864	1	2	yes	—
124	M₁₆	—	284	2	no	—
125	M₁₆	—	284	2	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	284	2	no	—
130	C₂×C₈	—	284	2	no	—
131	C₄⋊C₄	—	2	4	no	—
132	C₄⋊C₄	—	4	2	no	—
133	C₄⋊C₄	—	4	2	no	—
134	C₄²	C₂×D₂₈₄	1	1	yes	characteristic
135	C₄²	#11765864	1	2	yes	—
136	C₄²	#11765865	1	1	yes	characteristic
137	C₂³	#14556115	1	1	yes	characteristic
138	Q₈	—	4	4	no	—
139	Q₈	—	8	2	no	—
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	8	1	no	—
143	C₂×C₄	—	8	1	no	—
144	C₂×C₄	—	8	2	no	—
145	C₂×C₄	—	8	1	no	—
146	C₂×C₄	#14556102	1	1	yes	characteristic
147	C₂×C₄	#14556109	1	1	yes	characteristic
148	C₂×C₄	#14556112	1	2	yes	—
149	C₂×C₄	#14556121	1	2	yes	—
150	C₂×C₄	#14556148	1	2	yes	—
151	C₂×C₄	#14556154	1	2	yes	—
152	C₈	—	4	2	no	—
153	C₈	—	568	2	no	—
154	C₈	—	568	2	no	—
155	C₂²	—	2	2	no	—
156	C₂²	#17985924	1	1	yes	characteristic
157	C₂²	#17986353	1	1	yes	characteristic
158	C₂²	#17986363	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	2	2	no	—
161	C₄	—	8	2	no	—
162	C₄	—	16	1	no	—
163	C₄	#17986214	1	2	yes	—
164	C₄	#17986320	1	2	yes	—
165	C₂	—	2	1	no	—
166	C₂	—	2	1	no	—
167	C₂	#22589753	1	1	yes	characteristic
168	C₂	#22589804	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22588481

C₂

yes

maximal, characteristic

#22587012

C₂

yes

maximal

#22586198

C₂

yes

maximal

#22586185

C₂

yes

maximal, characteristic

C₇₁×#1960

C₂

yes

maximal, characteristic

#17986052

C₂²

yes

—

#17986023

C₂²

yes

—

#17985764

—

#17985745

—

C₂×C₄⋊Q₈×C₇₁

C₂²

yes

characteristic

C₂×#103×C₇₁

—

C₂×#97×C₇₁

C₂²

yes

characteristic

#382×C₇₁

C₂²

yes

characteristic

C₂×#11765849

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

—

C₂²⋊(C₇₁⋊C₈)

—

C₄×C₇₁⋊C₈

—

C₂²×Q₈×C₇₁

—

C₂²×C₈×C₇₁

D₄

yes

—

C₄⋊Q₈×C₇₁

—

C₄⋊Q₈×C₇₁

—

C₄⋊Q₈×C₇₁

D₄

yes

—

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄²×C₇₁

C₂³

yes

characteristic

#103×C₇₁

—

#97×C₇₁

—

#95×C₇₁

—

#11765849

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₁

#41

yes

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₄²×C₇₁

#41

yes

—

C₄²×C₇₁

yes

C₂³×C₇₁

yes

Q₈×C₇₁

—

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

—

C₂×C₄×C₇₁

#126

yes

—

C₂×C₄×C₇₁

#135

yes

—

C₂×C₄×C₇₁

AGL(1,ℤ/8)

yes

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#463

yes

—

C₄×C₇₁

#482

yes

—

#10635

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

#2073

yes

characteristic

C₂×C₇₁

#2082

yes

—

#1960

D₇₁

yes

—

—

—

—

C₇₁

#10635

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

C₂×D₇₁

yes

—

—

—

C₂×#97

C₂×D₇₁

yes

characteristic

#382

C₂×D₇₁

yes

characteristic

#341

—

142

—

#332

—

142

—

C₂²×Q₈

—

100

C₂×M₁₆

—

142

—

101

C₂²×C₈

C₂²⋊D₇₁

yes

—

102

C₄⋊Q₈

—

103

C₄⋊Q₈

—

104

C₄⋊Q₈

D₂₈₄

yes

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄²

C₂²×D₇₁

yes

characteristic

108

C₄⋊C₈

—

142

—

109

C₄⋊C₈

—

142

—

110

C₄⋊C₈

—

142

—

111

#103

—

112

C₂²⋊C₈

—

142

—

113

#97

—

114

C₄×C₈

—

142

—

115

#95

—

116

C₂×Q₈

—

117

C₂×Q₈

—

118

C₂×Q₈

—

119

C₂×Q₈

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

123

C₂²×C₄

#11765864

yes

—

124

M₁₆

—

284

—

125

M₁₆

—

284

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

284

—

130

C₂×C₈

—

284

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄²

C₂×D₂₈₄

yes

characteristic

135

C₄²

#11765864

yes

—

136

C₄²

#11765865

yes

characteristic

137

C₂³

#14556115

yes

characteristic

138

Q₈

—

139

Q₈

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

#14556102

yes

characteristic

147

C₂×C₄

#14556109

yes

characteristic

148

C₂×C₄

#14556112

yes

—

149

C₂×C₄

#14556121

yes

—

150

C₂×C₄

#14556148

yes

—

151

C₂×C₄

#14556154

yes

—

152

C₈

—

153

C₈

—

568

—

154

C₈

—

568

—

155

C₂²

—

156

C₂²

#17985924

yes

characteristic

157

C₂²

#17986353

yes

characteristic

158

C₂²

#17986363

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

#17986214

yes

—

164

C₄

#17986320

yes

—

165

C₂

—

166

C₂

—

167

C₂

#22589753

yes

characteristic

168

C₂

#22589804

yes

—

	subgroup	G/H
center	C₂²	#17985924
commutator, cosocle	C₂×C₄²×C₇₁	C₂³
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1960	C₂
socle	C₂²×C₇₁	#397
prime radical	C₂³×C₇₁	#124
hypercenter	#1960	D₇₁
hypocenter	C₇₁	#10635
LCS	G ▷ C₂×C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1960

subgroup

G/H

center

C₂²

#17985924

commutator, cosocle

C₂×C₄²×C₇₁

C₂³

Frattini

C₂×C₄²

C₂²×D₇₁

Fitting

C₇₁×#1960

C₂

C₂²×C₇₁

C₂³×C₇₁

D₇₁

C₇₁

G ▷ C₂×C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1960

N	H ≅ G/N	type
90. C₇₁	#10635	semidirect

H ≅ G/N

type

90. C₇₁

#10635

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22588481	2
#22587012	2
#22586198	2
#22586185	2
C₇₁×#1960	2
#10635	71

subgroup

index

C₇₁×#1960

Subgroups — #31902262

Conjugacy classes of subgroups

Complements

Maximal subgroups